What might cause a rash on my armpit?

Several causes. If the vault (the central "pit" area) is involved, it's usually caused by allergy to a deodorant or antiperspirant, or some component of it. If the outer rim of the armpit is involved and the vault is spared, it's usually caused by an article of clothing or detergent or fabric softener. Other causes include yeast and fungus infections, and there are other, more obscure causes.